The Ngazi Method of Exponential Function Calculation. Kuria Tony Kimani1 1Affiliation not available May 5, 2020 The Ngazi Method of Exponential Function Calculation Msc. Student, Research Methods Jomo Kenyatta University of Agriculture and Technology. Tel: +254-713 172 555 e-mail: [email protected] Abstract This paper presents an entirely new way of calculating exponential functions. This method uses the 2n sequence at its foundation. An example of the 2n sequence is 2,4,8,16,32,64 and so on. This method is vastly more time saving and energy saving because one performs very few multiplication operations once an exponent has been broken down into its 2n components. This method can be coded into a computer software and this will improve the speed with which computer libraries calculate exponential functions. This paper also explains how this new method of exponential function calculation can partly help to solve the discrete exponential and discrete logarithm problem through easier calculation of exponential functions. 1.Introduction. \Neural network simulations often spend a large proportion of their time computing exponential functions. Since the exponentiation routines of typical math libraries are rather slow, their replacement with a fast approximation can greatly reduce the overall computation time."[1] It is true that computer math libraries that deal with the exponential functions are rather slow but this is a function of using very archaic and long routine calculations which are time-consuming and energy wasting. If a more time-efficient method for calculating exponential functions can be found and then coded into a computer math library then this will save a lot of time for many mathematicians, engineers and computer scientists who use these exponential function libraries a lot. This author presents a new method of calculating exponential functions which is both accurate and time-saving. We no longer need to rely on time-saving approximations of exponential functions. We can now use time-saving accurate calculations of exponential functions and this could indeed change the field computational science. There is a relationship between exponential functions and logarithms. The power of logarithms as a compu- tational device lies in the fact that by them multiplication and division are reduced to the simpler operations Posted on Authorea 17 Jan 2020 | CC BY 4.0 | https://doi.org/10.22541/au.157928107.73534923 | This a preprint and has not been peer reviewed. Data may be preliminary. 1 of addition and subtraction.[2] \There are many applications of exponential functions and logarithmic functions in science and technology. The voltage in a given circuit can be expressed using exponents. The value of money in an investment can be determined through the use of exponents. The intensity of earthquakes is measured by a logarithmic scale. The intensity of light related to the thickness of the material through which it passes can be expressed using exponents. The distinction between acids and bases in chemistry is measured in terms of logarithms." [3] When it comes to exponential functions, the word exponent is often used instead of index, and functions in which the variable is in the index (such as 2x, 10sinx) are called exponential functions. [4] If b is a real number greater than zero, then for each real exponent x we assume bx is a unique real number. Since for each real x there is one and only one bx, the equation y=bx,(b>0) defines a function. We call such an equation an exponential function.[3] 2. Body . To solve the exponential function f(x)=mx, the most common method of calculating exponential functions has been to directly multiply m, x number of times. This method of calculation is extremely slow especially if the exponent has a large value. For, example calculating f(x) = 35123000 involves multiplying 35, 123,000 times. This is an extremely difficult task and is frankly almost impossible if calculated manually. Computers are best suited to calculate this function because computers do not get tired of performing repetitive tasks. However, there is a need to come up with a more efficient way of calculating the exponential function. This is important because exponential functions are very important in mathematics and engineering fields. This paper introduces a new, novel method of calculating the exponential function. This method is extremely efficient and it uses the 2n sequence at its foundation. One can solve the exponential function quicker if one uses the 2n table. A sample of the table will be displayed towards the end of the paper. I have decided to call this method the Ngazi method. Ngazi is a swahili word for ladder or stairs. Ngazi is a two syllable word which is pronounced as (nga-zi) where /n/ and /g/ are pronounced as one syllable.[5] I named this method ngazi or stairs because the results of the first multiplication are used in the second multiplication and the results of the second multiplication are used in the third multiplication and so on. Therefore the first multiplication is linked to the second multiplication and the second multiplication is linked to the third multiplication and this reminds the author of a flight of stairs where one stair leads to the next stair up to the final stair. 2.1 Introduction to the ngazi method Suppose you are trying to calculate the function f(x)= m16. The method you would use is probably the normal method we always use for calculating exponential func- tions. You would multiply m by itself 15 times to get m16. f(x)= m×m×m×m×m×m×m×m×m×m×m×m×m×m×m×m = m16. However, this method is tedious, time-consuming and inefficient. Now let us use the ngazi method to calculate this function. I will calculate it first, then I will explain after the calculation. Example 1 f(x)= m16 f(x)= m×m = m2. m2×m2 = m4 Posted on Authorea 17 Jan 2020 | CC BY 4.0 | https://doi.org/10.22541/au.157928107.73534923 | This a preprint and has not been peer reviewed. Data may be preliminary. 2 m4×m4 = m8 m8 ×m8 = m16 Notice that this new method exploits in its calculation, the fact that mn ×mn =m2n.. This is a well known property of the exponential function and in this paper, this property will be applied to all cases of exponential functions including functions that have exponents that do not belong to the 2n sequence. When two exponential functions are multiplied together, and the bases are similar, then we can add the exponents together. Therefore, instead of multiplying m by itself 15 times, we can multiply the results of every multiplication by itself and we will end up with only four distinct multiplication operations. Therefore we have reduced the number of multiplication operations from 15 to 4. We can calculate the percentage of reduction in inefficiency in multiplication operations. 15-4 = 11. Therefore, we have reduced the number of multiplication operations by 11. Therefore, in percentage terms we have reduced inefficient multiplication operations by (11÷15)×100= 73.33%. This is a 73.33% reduction in inefficient multiplication operations and it is a significant improvement in efficiency. What is magical about this method is the fact that the reduction in inefficiency actually increases as the value of the exponent keeps on increasing. I will give a second example with a slightly larger exponent value to prove this salient point. Example 2 Calculate f(x) = m256 Using the normal method f(x) = m×m×m×m×m×m×m×m×m×m×m×m×m×m×m×m m×m×m×m×m×m×m×m×m×m×m×m×m×m×m×m m×m×m×m×m×m×m×m×m×m×m×m×m×m×m×m m×m×m×m×m×m×m×m×m×m×m×m×m×m×m×m m×m×m×m×m×m×m×m×m×m×m×m×m×m×m×m m×m×m×m×m×m×m×m×m×m×m×m×m×m×m×m m×m×m×m×m×m×m×m×m×m×m×m×m×m×m×m m×m×m×m×m×m×m×m×m×m×m×m×m×m×m×m m×m×m×m×m×m×m×m×m×m×m×m×m×m×m×m m×m×m×m×m×m×m×m×m×m×m×m×m×m×m×m m×m×m×m×m×m×m×m×m×m×m×m×m×m×m×m m×m×m×m×m×m×m×m×m×m×m×m×m×m×m×m m×m×m×m×m×m×m×m×m×m×m×m×m×m×m×m m×m×m×m×m×m×m×m×m×m×m×m×m×m×m×m m×m×m×m×m×m×m×m×m×m×m×m×m×m×m×m m×m×m×m×m×m×m×m×m×m×m×m×m×m×m×m = m256. Notice that each row has 16 operands of m and we have a total of 16 rows therefore we have a total of 16×16 = 256 operands of m. We also have 255 (256-1) multiplication operations. To get the number of multiplication operations, subtract one from the total number of operands. 255 multiplication operations is unacceptably large and a more efficient method ought to be used instead. Let us use the ngazi method to calculate the function f(x) = m256 Posted on Authorea 17 Jan 2020 | CC BY 4.0 | https://doi.org/10.22541/au.157928107.73534923 | This a preprint and has not been peer reviewed.